Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

330 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsdifference of some parameter value characteristic to the systems, and the unperturbed kernelsT and C are taken at the parameter value a. Denoting this dependence in the formT(P,P') =T(P,P'|a)andC(P',P") =C(P',P"|a)the perturbed kernels readT(P,P') = T(P,P'|a + Aa)andC(P,P')= C(P,P'ja + Aa)Assuming a common contribution function of the form f(P,P'), we also postulate that thecontributions do not depend on the parameter a. This is the case, for example, with thetrack-length estimator if flux integrals are estimated, but is certainly not the case with theexpectation estimator. The considerations below will be extended to parameter-dependentcontributions in Section F.Let us consider score functions of the formF(s) = CiAas,i.e.,SW(Of(OW'(i)Aaf(i) (6.47)Obviously, by taking the limit of Aa —» 0, the argument of the score function tends to thederivative of the score, the quantity of interest in the differential game. We shall assumewithout further investigation that the order of taking the limit Aa •—» 0 and the expectationcan be interchanged, i.e., thatlim G(s) = G(Hm s)andIG( limAor—>0S 2lim M (6.48)We do not discuss this assumption, but, rather, note that for games that are feasible andhave a bounded variance, this assumption comes true at least for linear and quadratic functionof s in place of G(s). Note that the expectation in Equation (6.46) is taken with the unperturbed